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A Thurston boundary and visual sphere of the universal Teichmüller space

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Abstract

A Thurston boundary of the universal Teichmüller space T(ⅅ) is the space PMLbdd(ⅅ) of projective bounded measured laminations of ⅅ. A geodesic ray in T(ⅅ) is of generalized Teichmüller-type if it shrinks the vertical foliation of a holomorphic quadratic differential. We consider the limit points in PMLbdd(ⅅ) of certain geodesic rays induced by non-integrable holomorphic quadratic differentials. In particular, we show that there is a geodesic ray in T(ⅅ), such that the limiting projective measured lamination is supported on a geodesic lamination, whose leaves are not homotopic to leaves of either vertical or horizontal foliation of the corresponding holomorphic quadratic differential.

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References

  1. L. V. Ahlfors, Lectures on Quasiconformal Mappings, American Mathematical Society, Providence, RI, 2006.

    Book  Google Scholar 

  2. F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), 139–162.

    Article  MathSciNet  Google Scholar 

  3. F. Bonahon and D. Šarić, A Thurston boundary for infinite-dimensional Teichmüller spaces, Math. Ann., https://doi.org/10.1007/s00208-021-02148-z

  4. J. Brock, C. Leininger, B. Modami and K. Rafi, Limit sets of Weil—Petersson geodesics with nonminimal ending laminations, J. Topol. Anal. 12 (2020), 1–28.

    Article  MathSciNet  Google Scholar 

  5. J. Brock, C. Leininger, B. Modami and K. Rafi, Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical folliations, II, J. Reine Angew. Math. 758 (2020), 1–66.

    Article  MathSciNet  Google Scholar 

  6. J. Chaika, H. Masur and M. Wolf, Limits in PMF of Teichmüller geodesics, J. Reine Angew. Math. 747 (2019), 1–44.

    Article  MathSciNet  Google Scholar 

  7. F. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, American Mathematical Society, Providence, RI, 2000.

    MATH  Google Scholar 

  8. J. Garnett and D. Marshall, Harmonic Measure, Cambridge University Press, Cambridge, 2005.

    Book  Google Scholar 

  9. A. Fletcher and V. Markovic, Quasiconformal Maps and Teichmüller Theory, Oxford University Press, Oxford, 2007.

    MATH  Google Scholar 

  10. H. Hakobyan and D. Šarić, Vertical limits of graph domains, Proc. Amer. Math. Soc. 144 (2016), 1223–1234.

    Article  MathSciNet  Google Scholar 

  11. H. Hakobyan and D. Šarić, Limits of Teichmüller geodesics in the Universal Teichmüller space, Proc. Lond. Math. Soc. (3) 116 (2018), 1599–1628.

    Article  MathSciNet  Google Scholar 

  12. J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York, 2001.

    Book  Google Scholar 

  13. O. Lehto, Univalent Functions and Teichmüller Spaces, Springer, New York, 1987.

    Book  Google Scholar 

  14. O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, Springer, New York-Heidelberg, 1973.

    Book  Google Scholar 

  15. C. Leininger, A. Lenzhen and K. Rafi, Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliations, J. Reine Angew. Math. 737 (2018), 1–32.

    Article  MathSciNet  Google Scholar 

  16. A. Lenzhen, Teichmüller geodesics that do not have a limit in PMF, Geom. Topol. 12 (2008), 177–197.

    Article  MathSciNet  Google Scholar 

  17. H. Masur, Two boundaries of Teichmüller space, Duke Math. J. 49 (1982), 183–190.

    Article  MathSciNet  Google Scholar 

  18. H. Miyachi and D. Šarić, Uniform weak* topology and earthquakes in the hyperbolic plane, Proc. Lond. Math. Soc. (3) 105 (2012), 1123–1148.

    Article  MathSciNet  Google Scholar 

  19. D. Šarić, Geodesic currents and Teichmüller space, Topology 44 (2005), 99–130.

    Article  MathSciNet  Google Scholar 

  20. D. Šarić, Infinitesimal Liouville distributions for Teichmüller space, Proc. London Math. Soc. (3) 88 (2004), 436–454.

    Article  MathSciNet  Google Scholar 

  21. D. Šarić, A Thurston boundary for infinite-dimensional Teichmüller spaces: geodesic currents, preprint, arXiv:1505.01099 [math.GT].

  22. K. Strebel, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreiser, Comment. Math. Helv. 36 (1961/1962), 306–323.

    Article  MathSciNet  Google Scholar 

  23. K. Strebel, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreiser. II, Comment. Math. Helv. 39 (1964), 77–89.

    Article  MathSciNet  Google Scholar 

  24. J. Väisälä, Lectüres on n-Dimensional Quasiconformal Mappings, Springer, Berlin-New York, 1971.

    Book  Google Scholar 

Download references

Acknowledgement

We would like to thank the anonymous referee for carefully reading the paper and for the numerous comments which improved the paper and encouraged us to include Section 6.

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Authors and Affiliations

  1. Department of Mathematics, Kansas State University, Manhattan, KS, 66502, USA

    Hrant Hakobyan

  2. Department of Mathematics, Queens College of CUNY, 65-30 Kissena Blvd., Flushing, NY, 11367, USA

    Dragomir Šarić

  3. Mathematics PhD. Program, The CUNY Graduate Center, 365 Fifth Avenue, New York, NY, 10016-4309, USA

    Dragomir Šarić

Authors
  1. Hrant Hakobyan
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  2. Dragomir Šarić
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Corresponding author

Correspondence to Hrant Hakobyan.

Additional information

The second author was partially supported by National Science Foundation grant DMS 1102440 and by the Simons Foundation grant.

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Hakobyan, H., Šarić, D. A Thurston boundary and visual sphere of the universal Teichmüller space. JAMA 143, 681–721 (2021). https://doi.org/10.1007/s11854-021-0166-3

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  • DOI: https://doi.org/10.1007/s11854-021-0166-3

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